The code (314.29) says that J boxes must be accessible without removing part of the building. I don't know if cabinets constitute part of the building. It's also supposed to have its own cover. Your "workman" is a cheeseball, IMO. I can't imaging that you like the way this looks and I would have him fix it. If you are having a proper inspection, I doubt it would pass and that would be the final word.
I'm just a DIYer so keep listening to others. Good luck.

The exposed wires will definitely not pass inspection. Move the box either all the way in or all the way out of the cabinet.

"The code (314.29) says that J boxes must be accessible without removing part of the building."

This is not what the NEC says.

You cannot damage the finished surfaces in the building.

A box behind a screwed in place panel is acceptable, a nailed in place panel is not.

The vanity cabinet is not acceptable as an obstruction.

Either move the box from behind it and use a blank cover plate or put it completely behind the cabinet and cut an access hols in the back (if needed).

That is, in fact, almost precisely what the code says at 314.29. Later, I think in relation to metal boxes, it says, at 314.72 (D):
"Boxes shall be installed so that the wiring is accessible without removing any part of the building..."
There's no mention of "finished surfaces" although, in practice I would assume it's the same thing.
I do appreciate your expertise, but I CAN read. If I'm reading out of context, that would be helpful to know. Thanks.

My immediate recall agreed with that of brickeyee. It comes from the definitions in Article 100. Article 314.29 does not have the word finished.

About the second full sentence in the code is the definition of "accessible" in Chapter 1:

Accessible (as applied to wiring methods). Capable of being removed or exposed without damaging the building structure or finish or not permanently closed in by the structure or finish of the building.

Any way, regardless of which definition of accessible you are using, it means the same thing. Unless the vanity is an unattached piece of furniture (which due to plumbing they rarely are), it blocking a box does NOT meet the definition of accessible.

"That is, in fact, almost precisely what the code says at 314.29. Later, I think in relation to metal boxes, it says, at 314.72 (D):
"Boxes shall be installed so that the wiring is accessible without removing any part of the building..."
There's no mention of "finished surfaces" although, in practice I would assume it's the same thing.
I do appreciate your expertise, but I CAN read. If I'm reading out of context, that would be helpful to know. Thanks."

The NEC is lik any other legal document when it is adopted.

Words have their 'normal' meaning UNLESS they are otherwise defined.

The NEC defines what 'accessible' means rather explicitly in Article 100, Definitions.

IMO (and I'm not a AHJ), you'd probably be OK if you notched the vanity cabinet so that the receptacle is usable, and there is enough room to work on it reasonably well (unscrew it, remove it, rewire it).

That's a big notch and an ugly one.

I'd say your remuddlers need to put 'er in reverse and take another run at it, with the vanity a few inches to the right or left.

Or move the box.

However either move is going to require professionals which these clowns apparently are not.